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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Compan ies, Inc.

NAME DATE PERIOD

Lesson 2-6

Chapter 2 37 Glencoe Geometry

Algebraic Proof A list of algebraic steps to solve problems where each step is justified is called an algebraic proof

The table shows properties you have studied in algebra.The following properties are true for any real numbers a, b, and c.

Addition Property of Equality

If a = b, then a + c = b + c.

Subtraction Property of Equality

If a = b, the a - c = b - c.

Multiplication Property of Equality

If a = b, then a ? c = b

? c.

Division Property of Equality

If a = b and c ≠ 0, then,

a c b c

Reflexive Property of Equality

a = a

Symmetric Property of Equality

If a = b and b = a.

Transitive Property of Equality

If a = b and b = c, then a = c.

Substitution Property of Equality

If a = b,then a may be replaced by b in any equation or expression.

Distributive Property

a (b + c) = ab + ac Solve 6x + 2(x - 1) = 30. Write a justification for each step.

Algebraic Steps Properties

6x + 2(x - 1) = 30

Original equation or Given

6x + 2x - 2 = 30 Distributive Property

8x - 2 = 30 Substitution Property of Equality

8x - 2 + 2 = 30 + 2 Addition Property of Equality

8x = 32 Substitution Property of Equality

8 x 8 32
8

Division Property of Equality

x = 4 Substitution Property of Equality

Exercises

Complete each proof.

Study Guide and Intervention

Algebraic Proof

2-6

Example

2. Given:

4x + 8 =

x + 2

Prove:

x = -2

Proof:

Statements Reasons

a. 4x + 8 = x + 2 a. b. 4x + 8 - x = b. x + 2 - x c. 3x + 8 = 2 c. Substitution d. d. Subtr. Prop. e. e. Substitution f. 3x 3 6 3 f. g. g. Substitution 1. Given: 4x + 6 2 = 9

Prove:

x = 3

Proof:

Statements Reasons

a. 4 x + 6 2 = 9 a. b. 4 x + 6 2 = 2(9) b. Mult. Prop. c. 4x + 6 = 18 c. d. 4x + 6 - 6 = 18 - 6 d. e. 4x = e. Substitution f. 4x 4 = f. Div. Prop. g. g. Substitution Given 2 Subs.

Subtr. Prop.

12 x = 3Given

Subtr. Prop.

3x + 8 - 8 =

2 - 8 3 x = -6

Div. Prop.

x = -2 12 4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Compan ies, Inc.

NAME DATE PERIOD

Chapter 2 38 Glencoe Geometry

Geometric Proof Geometry deals with numbers as measures, so geometric proofs use properties of numbers. Here are some of the algebraic properties used in proofs. Write a two-column proof to verify this conjecture.

Given:

m?1 = m?2, m?2 = m?3

Prove: m?1 = m?3

Proof:

Exercises

State the property that justifies each statement.

1. If m?1 = m?2, then m?2 = m?1.

2. If m?1 = 90 and m?2 = m?1, then m?2 = 90.

3. If AB = RS and RS = WY, then AB = WY.

4. If AB = CD, then

1 2 AB = 1 2 CD.

5. If m?1 + m?2 = 110 and m?2 = m?3, then m?1 + m?3 = 110.

6. RS = RS

7. If AB = RS and TU = WY, then AB + TU = RS + WY.

8. If m?1 = m?2 and m?2 = m?3, then m?1 = m?3.

9. If the formula for the area of a triangle

is A = 1 2 bh, then bh is equal to 2 times the area of the triangle.

Write a two-column proof to verify

this conjecture. DAR T SCB 12 3

Study Guide and Intervention (continued)

Algebraic Proof

2-6

Property SegmentsAngles

Reflexive

AB = ABm? 1 = m? 1

Symmetric

If AB = CD, then CD = AB.If m? 1 = m? 2, then m? 2 = m? 1.

Transitive

If AB = CD and CD = EF, then AB = EF. If m? 1 = m? 2 and m? 2 = m? 3, then m? 1= m? 3.

Example

Statements Reasons

1. m?1 = m?2 2. m?2 = m?3 3. m?1 = m?31. Given 2. Given 3.

Transitive Property of Equality

Sym. Prop.

Subs.

Trans. Prop.

Mult. Prop.

Subs.

Refl. Prop.

Add. Prop.

Trans. Prop.

Given: A =

1 2 bh

Prove: 2

A = bh

Proof:

Statements Reasons

1. A = 1 2 bh 2.

2A = 2

1 2 bh

3. 2A = bh1.

Given 2.

Mult. Property

3. Subs.quotesdbs_dbs14.pdfusesText_20

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